Open string topological amplitudes and gaugino masses

TL;DR

This work studies moduli-dependent couplings of higher-derivative F-terms $(\mathrm{Tr} W^2)^{h-1}$ in Type I string theory with open strings, showing they are governed by the genus-zero topological partition function $F^{(0,h)}$ on bordered worldsheets and are related to heterotic duals. It demonstrates a holomorphic anomaly that introduces $\Pi$-terms, leading to R-symmetry breaking for $h\ge3$ and a SUSY-breaking-induced gaugino mass $m_{1/2}$ that scales as $m_0^4$ in string units, with Dirac masses for non-chiral brane fermions from $\Pi\mathrm{Tr}W^2$. The paper provides explicit calculations in magnetized D9-brane toroidal models, deriving closed-form expressions for $F^{(0,3)}$ and the related $F^{(0,2)}_{\bar i;\bar j}$ contributions, and demonstrates how these topological amplitudes map to low-energy spectra and mass terms. These results illuminate a string-theoretic mechanism to generate gaugino masses and fermion masses in scenarios with high scalar masses, with potential relevance to split SUSY and brane-intersection setups.

Abstract

We discuss the moduli-dependent couplings of the higher derivative F-terms $(\Tr W^2)^{h-1}$, where $W$ is the gauge N=1 chiral superfield. They are determined by the genus zero topological partition function $F^{(0,h)}$, on a world-sheet with $h$ boundaries. By string duality, these terms are also related to heterotic topological amplitudes studied in the past, with the topological twist applied only in the left-moving supersymmetric sector of the internal $N=(2,0)$ superconformal field theory. The holomorphic anomaly of these couplings relates them to terms of the form $Π^n({\rm Tr}W^2)^{h-2}$, where $Π$'s represent chiral projections of non-holomorphic functions of chiral superfields. An important property of these couplings is that they violate R-symmetry for $h\ge 3$. As a result, once supersymmetry is broken by D-term expectation values, $(\Tr W^2)^2$ generates gaugino masses that can be hierarchically smaller than the scalar masses, behaving as $m_{1/2}\sim m_0^4$ in string units. Similarly, $Π{\rm Tr}W^2$ generates Dirac masses for non-chiral brane fermions, of the same order of magnitude. This mechanism can be used for instance to obtain fermion masses at the TeV scale for scalar masses as high as $m_0\sim{\cal O}(10^{13})$ GeV. We present explicit examples in toroidal string compactifications with intersecting D-branes.

Figures (6)

• Figure 1: Open string world-sheet with $h$ boundaries.
• Figure 2: The involution (\ref{['invh']}) acts on the $g=h{-}1$ double-cover as a mirror reflection in the horizontal plane. The open string world-sheet is in the upper half while its mirror image is in the shaded lower half.
• Figure 3: The fundamental polygon obtained by cutting the open string world-sheet along the curves shown in Figure 1.
• Figure 4: The three degeneration limits.
• Figure 5: The $(g=0, h=3)$ surface $\Sigma_{(0,3)}$ is in the upper half of its $g=2$ double-cover. ${\bf c}$ is the dividing geodesics.